Here, we consider the on-site interactions with \(V_{0}\)  denoting the
effective interaction potential [1]}, [2]}. By making
the mean-field decoupling, \(H_{P}\)  can be rewritten in terms of the
SC pairings as,
\(H_{P}&=&\sum _{i\in A}(\Delta _{A}a^{\dag }_{i,\uparrow }a^{\dag }_{i,\downarrow }+h.c.)+\sum _{i\in B}(\Delta _{B}b^{\dag }_{i,\uparrow }b^{\dag }_{i,\downarrow } \nonumber \\&&+h.c.),\) 
